Calculating the Shortest Line between two Lines

Developer: RhinoScript
Summary: Demonstrates how to calculate the shortest line between two lines.

Two lines in three dimensions generally do not intersect at a point. They may be parallel (no intersections) or they may be coincident (infinite intersections). But, most often only their projection onto a plane intersects. When they do not exactly intersect at a point they can be connected by a line segment, the shortest line segment is unique and is often considered to be their intersection in 3-D.

The following example code demonstrates how to calculate the shortest line between two line segments using RhinoScript.

 Option Explicit
 
 ' Description:
 '   Returns the shortest line segment between 
 '   two infinite line segments.
 ' Parameters:
 '   p1 - the starting point of the first line
 '   p2 - the ending point of the first line
 '   p3 - the starting point of the second line
 '   p4 - the ending point of the second line
 ' Returns 
 '   Array - the shortest line segment if successful
 '   Null - if not successful or on error

 Function LineLineIntersect(p1, p2, p3, p4)
 
   LineLineIntersect = Null
 
   Const EPS = 2.2204460492503131e-016  
   Dim p13(2), p43(2), p21(2)
   Dim d1343, d4321, d1321, d4343, d2121, numer, denom, mua, mub
   Dim pa(2), pb(2)
 
   p13(0) = p1(0) - p3(0)
   p13(1) = p1(1) - p3(1)
   p13(2) = p1(2) - p3(2)
   p43(0) = p4(0) - p3(0)
   p43(1) = p4(1) - p3(1)
   p43(2) = p4(2) - p3(2)
 
   If Abs(p43(0)) < EPS And Abs(p43(1)) < EPS And Abs(p43(2)) < EPS Then Exit Function
 
   p21(0) = p2(0) - p1(0)
   p21(1) = p2(1) - p1(1)
   p21(2) = p2(2) - p1(2)
 
   If Abs(p21(0)) < EPS And Abs(p21(1)) < EPS And Abs(p21(2)) < EPS Then Exit Function
 
   d1343 = p13(0) * p43(0) + p13(1) * p43(1) + p13(2) * p43(2)
   d4321 = p43(0) * p21(0) + p43(1) * p21(1) + p43(2) * p21(2)
   d1321 = p13(0) * p21(0) + p13(1) * p21(1) + p13(2) * p21(2)
   d4343 = p43(0) * p43(0) + p43(1) * p43(1) + p43(2) * p43(2)
   d2121 = p21(0) * p21(0) + p21(1) * p21(1) + p21(2) * p21(2)
 
   denom = d2121 * d4343 - d4321 * d4321
   If Abs(denom) < EPS Then Exit Function
 
   numer = d1343 * d4321 - d1321 * d4343
   mua = numer / denom
   mub = (d1343 + d4321 * mua) / d4343
 
   pa(0) = p1(0) + mua * p21(0)
   pa(1) = p1(1) + mua * p21(1)
   pa(2) = p1(2) + mua * p21(2)
   pb(0) = p3(0) + mub * p43(0)
   pb(1) = p3(1) + mub * p43(1)
   pb(2) = p3(2) + mub * p43(2)
 
   LineLineIntersect = Array(pa, pb)
 
 End Function

You can test the above function as follows:

 Sub Test
 
   Dim l0 : l0 = Rhino.GetObject("Select first line", 4)
   Dim l1 : l1 = Rhino.GetObject("Select second line", 4)
 
   Dim p1 : p1 = Rhino.CurveStartPoint(l0)
   Dim p2 : p2 = Rhino.CurveEndPoint(l0)
   Dim p3 : p3 = Rhino.CurveStartPoint(l1)
   Dim p4 : p4 = Rhino.CurveEndPoint(l1)
 
   Dim rc : rc = LineLineIntersect(p1, p2, p3, p4)
   If IsArray(rc) Then
     Rhino.AddPoints rc
   End If
 
 End Sub